MHB How Do You Convert Temperatures and Solve Inverse Functions?

AI Thread Summary
Temperatures can be converted from Fahrenheit to Celsius using the function f(x) = 5/9(x − 32). The calculation for f(59) results in 15 degrees Celsius. To find the inverse function f^(-1)(x), one must rearrange the equation y = 5/9(x − 32) to express x in terms of y. Additionally, the set K, defined as {x : f(x) = x}, requires solving the equation 5/9(x − 32) = x to identify its elements. This discussion emphasizes the importance of understanding function inverses and solving equations in temperature conversion.
charlottecain
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Temperatures can be converted from Fahrenheit to Celsius using the
function f(x) = 5
/9
(x − 32).
(a) Calculate f(59).
(b) Find f
−1
(x), and verify that f
−1
(f(59)) = 59.
(c) Let K be the set {x : f(x) = x}. Find all elements of K and list K
 
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Hi, and welcome to the forum!
charlottecain said:
Temperatures can be converted from Fahrenheit to Celsius using the
function $f(x) = \frac59(x − 32)$.
(a) Calculate $f(59)$.
\[
f(59)=\frac59(59-32)=\frac59\cdot27=5\cdot\frac{27}{9}=5\cdot 3=15.
\]

charlottecain said:
(b) Find $f^{-1}(x)$, and verify that $f^{-1}(f(59)) = 59$.
To find the inverse of $f$ you need to solve the equation $y=\frac59(x − 32)$ for $x$, i.e., express $x$ through $y$. Can you do this? Start by multiplying both sides by $\frac95$.

charlottecain said:
(c) Let K be the set {x : f(x) = x}. Find all elements of K and list K
To do this you need to solve the equation $\frac59(x − 32)=x$. Can you do this?

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